Question

For each of the following matrices, determine whether the columns of the matrix are linearly independent or linearly dependent. if they dependent, give a non-zero vector X such that AX=0

A= [1 -3 -6]
[1 -3 -6]
[2 -5 -10]
answer______

Answer 1

The columns of the given matrix are linearly dependent because each column is a multiple of the others. A confirming example is the non-zero vector X=[1, 1, 1]T, which satisfies AX=0, proving that the columns are not linearly independent.

Step-by-step explanation:

To determine if the columns of a matrix are linearly independent or linearly dependent, we look for a non-zero vector X such that AX = 0, where A is the matrix in question. The matrix provided is:

A = \[\begin{pmatrix} 1 & -3 & -6 \\ 1 & -3 & -6 \\ 2 & -5 & -10\end{pmatrix}\]

By inspection, we can see that each column is a multiple of the others, indicating that the columns are not linearly independent. To show this, we can find a vector X such that AX = 0.

Let's take the first and second columns:\[X = \begin{pmatrix} 1 \\ -1 \\ 0\end{pmatrix}\]This indicates that the first column minus the second column yields the zero vector, thus demonstrating that the columns are linearly dependent. Another vector that satisfies AX = 0 for all columns simultaneously can be:\[X = \begin{pmatrix} 1 \\ 1 \\ 1\end{pmatrix}\]In this case, each column multiplied by 1 and added together would give the zero vector, again proving linear dependence.